Trembling hand perfect equilibrium

First define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy in an n-player strategic game where every pure strategy is played with positive probability. This is the "trembling hands" of the players; they sometimes play a different strategy, other than the one they intended to play. Then define a mixed strategy profile \sigma=(\sigma_1,\ldots,\sigma_n) as being trembling hand perfect if there is a sequence of perturbed games strategy profiles \{\sigma^k\}_{k=1,2,\ldots} that converges to \sigma such that for every k and every player 1\leq i \leq n the strategy \sigma_i is the best reply to \sigma^k_{-i}. Note: All completely mixed Nash equilibria are perfect. Note 2: The mixed strategy extension of any finite normal-form game has at least one perfect equilibrium. == Example ==

The game represented in the following normal form matrix has two pure strategies Nash equilibria, namely \langle \text{Up}, \text{Left}\rangle and \langle \text{Down}, \text{Right}\rangle. However, only \langle \text{U},\text{L}\rangle is trembling-hand perfect. Assume player 1 (the row player) is playing a mixed strategy (1-\varepsilon, \varepsilon), for 0. Player 2's expected payoff from playing L is: :1(1-\varepsilon) + 2\varepsilon = 1+\varepsilon Player 2's expected payoff from playing the strategy R is: :0(1-\varepsilon) + 2\varepsilon = 2\varepsilon For small values of \varepsilon, player 2 maximizes his expected payoff by placing a minimal weight on R and a maximal weight on L. By symmetry, player 1 should place a minimal weight on D and a maximal weight on U if player 2 is playing the mixed strategy (1-\varepsilon, \varepsilon). Hence \langle \text{U},\text{L}\rangle is trembling-hand perfect. However, a similar analysis fails for the strategy profile \langle \text{D}, \text{R}\rangle. Assume player 2 is playing a mixed strategy (\varepsilon, 1-\varepsilon). Player 1's expected payoff from playing U is: :1\varepsilon + 2(1-\varepsilon) = 2-\varepsilon Player 1's expected payoff from playing D is: :0\varepsilon + 2(1-\varepsilon) = 2-2\varepsilon For all positive values of \varepsilon, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence \langle \text{D}, \text{R}\rangle is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1. == Equilibria of two-player games ==

For 2×2 games, the set of trembling-hand perfect equilibria coincides with the set of equilibria consisting of two undominated strategies. In the example above, we see that the equilibrium Down,Right> is imperfect, as Left (weakly) dominates Right for Player 2 and Up (weakly) dominates Down for Player 1. == Equilibria of extensive form games ==

There are two possible ways of extending the definition of trembling hand perfection to extensive form games. • One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed games, every strategy of the extensive-form game must be played with non-zero probability. This leads to the notion of a normal-form trembling hand perfect equilibrium. • Alternatively, one may recall that trembles are to be interpreted as modelling mistakes made by the players with some negligible probability when the game is played. Such a mistake would most likely consist of a player making another move than the one intended at some point during play. It would hardly consist of the player choosing another strategy than intended, i.e. a wrong plan for playing the entire game. To capture this, one may define the perturbed game by requiring that every move at every information set is taken with non-zero probability. Limits of equilibria of such perturbed games as the tremble probabilities go to zero are called extensive-form trembling hand perfect equilibria. The notions of normal-form and extensive-form trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensive-form game may be normal-form trembling hand perfect but not extensive-form trembling hand perfect and vice versa. As an extreme example of this, Jean-François Mertens has given an example of a two-player extensive form game where no extensive-form trembling hand perfect equilibrium is admissible, i.e., the sets of extensive-form and normal-form trembling hand perfect equilibria for this game are disjoint. An extensive-form trembling hand perfect equilibrium is also a sequential equilibrium. A normal-form trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so. In fact, a normal-form trembling hand perfect equilibrium does not even have to be subgame perfect. == Problems with perfection ==

Myerson (1978) pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium. == References ==